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Theorem erdszelem10 27018
Description: Lemma for erdsze 27020. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
erdszelem.r  |-  ( ph  ->  R  e.  NN )
erdszelem.s  |-  ( ph  ->  S  e.  NN )
erdszelem.m  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
Assertion
Ref Expression
erdszelem10  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Distinct variable groups:    x, y    m, n, x, y, F   
n, I, x, y   
n, J, x, y    R, m, x, y    m, N, n, x, y    ph, m, n, x, y    S, m, x, y    T, m
Allowed substitution hints:    R( n)    S( n)    T( x, y, n)    I( m)    J( m)

Proof of Theorem erdszelem10
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 fzfi 11790 . . . . . . . 8  |-  ( 1 ... ( R  - 
1 ) )  e. 
Fin
2 fzfi 11790 . . . . . . . 8  |-  ( 1 ... ( S  - 
1 ) )  e. 
Fin
3 xpfi 7579 . . . . . . . 8  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin )
41, 2, 3mp2an 667 . . . . . . 7  |-  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  e. 
Fin
5 ssdomg 7351 . . . . . . 7  |-  ( ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  e.  Fin  ->  ( ran  T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
64, 5ax-mp 5 . . . . . 6  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
7 domnsym 7433 . . . . . 6  |-  ( ran 
T  ~<_  ( ( 1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
86, 7syl 16 . . . . 5  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
9 erdszelem.m . . . . . . . 8  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
10 hashxp 12192 . . . . . . . . . 10  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) ) )
111, 2, 10mp2an 667 . . . . . . . . 9  |-  ( # `  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )
12 erdszelem.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
13 nnm1nn0 10617 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
14 hashfz1 12113 . . . . . . . . . . 11  |-  ( ( R  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( R  -  1 ) ) )  =  ( R  -  1 ) )
1512, 13, 143syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( R  - 
1 ) ) )  =  ( R  - 
1 ) )
16 erdszelem.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
17 nnm1nn0 10617 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
18 hashfz1 12113 . . . . . . . . . . 11  |-  ( ( S  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( S  -  1 ) ) )  =  ( S  -  1 ) )
1916, 17, 183syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( S  - 
1 ) ) )  =  ( S  - 
1 ) )
2015, 19oveq12d 6108 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... ( R  - 
1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
2111, 20syl5eq 2485 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
22 erdsze.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN )
2322nnnn0d 10632 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
24 hashfz1 12113 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2523, 24syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
269, 21, 253brtr4d 4319 . . . . . . 7  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) ) )
27 fzfid 11791 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
28 hashsdom 12140 . . . . . . . 8  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin  /\  (
1 ... N )  e. 
Fin )  ->  (
( # `  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) )  <  ( # `  (
1 ... N ) )  <-> 
( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) ) )
294, 27, 28sylancr 658 . . . . . . 7  |-  ( ph  ->  ( ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) )  <->  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ~<  (
1 ... N ) ) )
3026, 29mpbid 210 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) )
31 erdsze.f . . . . . . . 8  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
32 erdszelem.i . . . . . . . 8  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
33 erdszelem.j . . . . . . . 8  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
34 erdszelem.t . . . . . . . 8  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
3522, 31, 32, 33, 34erdszelem9 27017 . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
36 f1f1orn 5649 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T : ( 1 ... N ) -1-1-onto-> ran  T )
37 ovex 6115 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
3837f1oen 7326 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ran  T  ->  (
1 ... N )  ~~  ran  T )
3935, 36, 383syl 20 . . . . . 6  |-  ( ph  ->  ( 1 ... N
)  ~~  ran  T )
40 sdomentr 7441 . . . . . 6  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
)  /\  ( 1 ... N )  ~~  ran  T )  ->  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
4130, 39, 40syl2anc 656 . . . . 5  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
428, 41nsyl3 119 . . . 4  |-  ( ph  ->  -.  ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
43 nss 3411 . . . . 5  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s
( s  e.  ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
44 df-rex 2719 . . . . 5  |-  ( E. s  e.  ran  T  -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. s ( s  e. 
ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
4543, 44bitr4i 252 . . . 4  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
4642, 45sylib 196 . . 3  |-  ( ph  ->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
47 f1fn 5604 . . . 4  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T  Fn  ( 1 ... N ) )
48 eleq1 2501 . . . . . 6  |-  ( s  =  ( T `  m )  ->  (
s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) ) )
4948notbid 294 . . . . 5  |-  ( s  =  ( T `  m )  ->  ( -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5049rexrn 5842 . . . 4  |-  ( T  Fn  ( 1 ... N )  ->  ( E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5135, 47, 503syl 20 . . 3  |-  ( ph  ->  ( E. s  e. 
ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5246, 51mpbid 210 . 2  |-  ( ph  ->  E. m  e.  ( 1 ... N )  -.  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
53 fveq2 5688 . . . . . . . . . 10  |-  ( n  =  m  ->  (
I `  n )  =  ( I `  m ) )
54 fveq2 5688 . . . . . . . . . 10  |-  ( n  =  m  ->  ( J `  n )  =  ( J `  m ) )
5553, 54opeq12d 4064 . . . . . . . . 9  |-  ( n  =  m  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  m ) ,  ( J `  m ) >. )
56 opex 4553 . . . . . . . . 9  |-  <. (
I `  m ) ,  ( J `  m ) >.  e.  _V
5755, 34, 56fvmpt 5771 . . . . . . . 8  |-  ( m  e.  ( 1 ... N )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5857adantl 463 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5958eleq1d 2507 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  <. ( I `
 m ) ,  ( J `  m
) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
60 opelxp 4865 . . . . . 6  |-  ( <.
( I `  m
) ,  ( J `
 m ) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <-> 
( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6159, 60syl6bb 261 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6261notbid 294 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  -.  (
( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
63 ianor 485 . . . 4  |-  ( -.  ( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6462, 63syl6bb 261 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  \/ 
-.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6564rexbidva 2730 . 2  |-  ( ph  ->  ( E. m  e.  ( 1 ... N
)  -.  ( T `
 m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N ) ( -.  ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6652, 65mpbid 210 1  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   E.wrex 2714   {crab 2717    C_ wss 3325   ~Pcpw 3857   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   `'ccnv 4835   ran crn 4837    |` cres 4838   "cima 4839    Fn wfn 5410   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415    Isom wiso 5416  (class class class)co 6090    ~~ cen 7303    ~<_ cdom 7304    ~< csdm 7305   Fincfn 7306   supcsup 7686   RRcr 9277   1c1 9279    x. cmul 9283    < clt 9414    - cmin 9591   NNcn 10318   NN0cn0 10575   ...cfz 11433   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100
This theorem is referenced by:  erdszelem11  27019
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